On this website, there are many questions about books on probability theory, but I would like to ask if you can suggest a book (or more than one if necessary) that is:
- rigorous and accurate according to modern standards;
- complete: from basic concepts and ideas to really advanced material;
- rich of useful applications of techniques and ideas of probability theory to other branches of mathematics and physics.
Best Answer
In my opinion, three books that meet the criteria listed in the OP are:
Probability Theory: The Logic of Science by E.T. Jaynes, 2002 version: this is a comprehensive book that illustrates probability theory from basic (first volume) to advanced concepts (second volume), combining a modern, rigorous approach with an enjoying style, often using interesting digressions that go beyond conventional mathematics, and analyzing a large number of applications of probability in physics, mechanics, and other fields. In my opinion, this probability textbook is surely one of the best choice for readers interested in applications and ideas, and who look for something more than conventional statistical books.
Applied probability by K. Lange, 2010: a very interesting textbook with a particolar focus on a variety of applications and examples taken from biology and computational techniques, written with a rigorous mathematical style. It starts from elementary probability theories and covers a large variety of applications of measure theory, cominatorics, expectations, optimization theory, and so on, extending to more advanced issues such as modern probabilistic approach to steady flow, Brownian motion, dynamics of porous media, population modeling, and many others. A detailed analysis of stochastic processes including Poisson and branching processes, Markov chains, martingales, and diffusion processes is also provided in a very clear manner and with several practical examples. In summary, this is another textbook suitable for those who search something "more" in the applications of probability theory.
Probability in Physics by Y. Ben-Menaheml and M. Hemmo, 2012: this is a nice high-level book included in the Springer "Frontier collection", a series of volumes dedicated to advanced, open issues at the forefront of modern science. The book is very well-structured, written in a readable and accessible manner, and covers a large range of issues and ideas of probability in physics, starting from very basic concepts such as notion of probability and chance in physics, tipicality, role of Lebesgue measures in statistic mechanics, and so on, and passing through a number of applications that include axioms of quantum mechanics, contemporary probabilistic approach to equilibrium, controversial relation between locality and determinism, modern view of Maxwell's demons, quantum decoherence, and many other. In synthesis, a third book that may be of interest for those who search mathematical rigour, applications and ideas.